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Full text of "Treatise On Analysis Vol-Ii"

286       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

denote the weak topology on Mn(G) corresponding to the vector space E5/2; the
topology ^"5/2 is finer than ^"2 and coarser than ^~3. Give an example of a se-
quence (ju,fl) which tends to 0 with respect to ^"5/2, and a sequence (/) of functions in
&l(G, ft) such that the sequence (/  ft) converges to 0 with respect to ^~6, but such
that the sequence (fj,n * (/ - ft)) does not tend to 0 with respect to ^~3 - (Take G = R
and/,(0 to be the function which is equal to sin nt in the interval [0, TT] and is zero
elsewhere.)

(c)    Let (/zfl) be a sequence of bounded measures on G which converges to ft with
respect to ys/2, let/e JP^G, ft) and let (/) be a sequence of functions in &l(G, ft)
such that Ni(/ /n)->0. Show that NI(/-&*/  //,*/) -> 0. (Reduce to the case
where ft = 0 and/n =/for all n, and then to the case where fe Jf(G). Show that the
sequence (ftn * (/ ft)) tends to 0 with respect to ^"3, by remarking that if g is a bounded
continuous function on G, the function/* g is uniformly continuous with respect to a
left-invariant distance on G.) Show that the result is no longer valid if ^~5/2 is replaced

ty*V

(d)    Let (//.) be a sequence of bounded measures on G which tends to 0 with respect
to ^"s/2, and (vn) a sequence of bounded measures on G which tends to v with respect
to ^2  Show that the sequence (ftn * vn) tends to 0 with respect to ^"5/2  (By using
(14.11.1), reduce to proving that </*#, ju,n * vn>->0 for any/e ^f(G) and g uni-
formly continuous with respect to a left-invariant distance on G. Then use (c).)

6.   The notation is that of Problem 5.

(a)    Let (/v) be a sequence of bounded measures on G. Suppose that, for each function
/e J&?HG, f$)9 the sequence (^n * (/ ft)) tends to 0 with respect to ^"2. Show that the
sequence of norms (||/xn||) is bounded. (Apply the Banach-Steinhaus theorem to the
sequence of mappings /h->(/xn */)" of LJ(G, fi) into U(G, ft), and use Problem 2(b).)
Deduce that the sequence (/v) tends to 0 with respect to ^"2.

(b)    Suppose that, for each/e ^fx(G, ft), the sequence ((! * (/ ft)) converges vaguely
to 0. Show that the sequence (/) tends vaguely to 0. (If K is any compact subset of G,
show as in (a) that the sequence (|/x,n|(K)) is bounded.) Give an example in which the
sequence (]|f&B||) is not bounded (take G = Z).

(c)    Suppose that, for each fe &l(G, ft), the sequence (//, * (/ ft)) converges to 0
with respect to ^5/2. Show that ju,n -> 0 with respect to 5"5/2 (use (a)). Give an example
where N^/z,, */)-0 for each /e &l(G, ft) but the sequence (jun) does not tend to 0
with respect to ^"3.

10. CONVOLUTION OF TWO FUNCTIONS

(14.10.1) Let f and g be two (complex) locally p-integrable functions on G.
Then the measures f- /? and g - f) are convolvable if and only if there exists a
^-negligible set N such that the function st-*g(s~lx)f(s) is $-integrable for all
x $ N and the function

r

/c~ lv\ //,,M    >7/?/'r,\res (/An * (/  j8)) converges to (/u,B * (/  j8)) with respect to ^6